Geometric Function Theory and Mathematical Physics

几何函数论与数学物理

• 批准号: RGPIN-2019-04940
• 负责人: Binder, Ilia
• 金额: $ 1.89万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=737960
• 关键词: Geometric; Function; Theory; Mathematical; Physics

In recent years, the classical methods of the Geometric Function Theory found numerous applications in the various areas of Mathematical Physics. The proposed research will concentrate on two of these areas. The first one is the Schramm Loewner Evolution (SLE) and its relation to the critical lattice models of Statistical Physics. The proposed research will lead to a better understanding of the fine geometric properties of the SLE curves. One of our aims is to rigorously confirm empirical predictions on the integral mixed spectrum, a major tool that describes the precise behaviour of the harmonic measure and boundary rotation. Other problems in this area are the conformal welding for SLE curves and the Conformal Dimension of the SLE curves, the smallest Hausdorff dimension of their quasiconformal images.  SLE was initially introduced as a conjectural scaling limit of certain interfaces for the lattice models of Statistical Physics. In a few situations, such as Percolation, Ising model, Gaussian Free Field, and Loop-Erased Random Walk, this conjecture has been established. I intend to develop a general approach to the rate of convergence results for a wide class of lattice models. This would imply, in particular, that the convergence for the critical Percolation and the Ising model is polynomial in the size of the lattice. I will also look for new observables to establish convergence for other Lattice Models.  This study can be viewed as a qualitative theory of critical phenomenon. The second area I will work on is the spectral theory of quasi-periodic Schroedinger and Jacobi operators.  I will investigate the Inverse Spectral Problem, one of the central questions of this extremely active field of Mathematical Physics. An important ingredient of our proposed research in this direction is the study of the fine properties of the harmonic measure for planar comb domains. These methods can also be used in the applications of the spectral theory to the KdV equation. Recently, D. Damanik, M. Goldstein, M. Lukic and I were able to verify a particular case of Deift's conjecture on the existence, uniqueness and almost periodicity in time of the solutions of KdV with almost periodic initial conditions. I plan to study the further interplay between quasi-periodicity and integrability. Applications of Geometric Function Theory frequently involve numerical simulations. Understanding the computational complexity of the related problems is thus essential. I will investigate the computability properties of the Green functions and Green maps as well as the Koebe maps and conformal weldings. I also plan to continue working on the fundamental theoretical questions of the Geometric Function Theory, such as Brennan's conjecture and related topics.

近年来，几何函数论的经典方法在数学物理的各个领域得到了广泛的应用。拟议的研究将集中在其中两个领域。第一部分是Schramm Loewner演化（SLE）及其与统计物理临界晶格模型的关系，这将有助于更好地理解SLE曲线的精细几何性质。我们的目标之一是严格确认经验预测的积分混合谱，一个主要的工具，描述了精确的行为的调和措施和边界旋转。这一领域的其他问题是SLE曲线的保形焊接和SLE曲线的保形维数，即它们的拟共形像的最小Hausdorff维数。SLE最初是作为统计物理学格子模型中某些界面的几何标度极限而引入的。在一些情况下，如渗流，伊辛模型，高斯自由场，环擦除随机游动，这个猜想已经成立。我打算发展一个一般的方法来计算收敛速度的结果，为广泛的一类格模型。这将意味着，特别是，临界渗流和伊辛模型的收敛是多项式的大小的晶格。我也将寻找新的观测量来建立其他格点模型的收敛性。这项研究可以被看作是临界现象的定性理论。第二个领域是准周期薛定谔和雅可比算子的谱理论，我将研究逆谱问题，这是数学物理学这一极其活跃领域的中心问题之一。在这个方向上，我们提出的研究的一个重要组成部分是研究平面梳状域的调和测度的优良性质。这些方法也可用于谱理论在KdV方程中的应用。最近，D.达马尼克湾戈尔茨坦，M。Lukic和我能够验证一个特殊的情况下Deift猜想的存在性，唯一性和几乎周期性的时间解决方案的KdV几乎周期性的初始条件。我计划进一步研究准周期性和可积性之间的相互作用。几何函数论的应用经常涉及数值模拟。因此，理解相关问题的计算复杂性是至关重要的。我将研究绿色函数和绿色映射以及Koebe映射和保形焊接的可计算性。我还计划继续研究几何函数论的基本理论问题，如布伦南猜想和相关主题。